Optimal. Leaf size=31 \[ \frac {\tan ^5(a+b x)}{5 b}+\frac {\tan ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2607, 14} \[ \frac {\tan ^5(a+b x)}{5 b}+\frac {\tan ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rubi steps
\begin {align*} \int \sec ^4(a+b x) \tan ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\tan ^3(a+b x)}{3 b}+\frac {\tan ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 1.81 \[ -\frac {2 \tan (a+b x)}{15 b}+\frac {\tan (a+b x) \sec ^4(a+b x)}{5 b}-\frac {\tan (a+b x) \sec ^2(a+b x)}{15 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 39, normalized size = 1.26 \[ -\frac {{\left (2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} - 3\right )} \sin \left (b x + a\right )}{15 \, b \cos \left (b x + a\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 26, normalized size = 0.84 \[ \frac {3 \, \tan \left (b x + a\right )^{5} + 5 \, \tan \left (b x + a\right )^{3}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 1.35 \[ \frac {\frac {\sin ^{3}\left (b x +a \right )}{5 \cos \left (b x +a \right )^{5}}+\frac {2 \left (\sin ^{3}\left (b x +a \right )\right )}{15 \cos \left (b x +a \right )^{3}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 26, normalized size = 0.84 \[ \frac {3 \, \tan \left (b x + a\right )^{5} + 5 \, \tan \left (b x + a\right )^{3}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 25, normalized size = 0.81 \[ \frac {{\mathrm {tan}\left (a+b\,x\right )}^3\,\left (3\,{\mathrm {tan}\left (a+b\,x\right )}^2+5\right )}{15\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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